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The goal of this paper is to investigate a new model, called generalized polynomial complementarity problems over a polyhedral cone and denoted by GPCPs, which is a natural extension of the polynomial complementarity problems and generalized tensor complementarity problems. Firstly, the properties of the set of all $$R^{K}_{{\varvec{0}}}$$ -tensors are investigated. Then, the nonemptiness and compactness of the solution set of GPCPs are proved, when the involved tensor in the leading term of the polynomial is an $$ER^{K}$$ -tensor. Subsequently, under fairly mild assumptions, lower bounds of solution set via an equivalent form are obtained. Finally, a local error bound of the considered problem is derived. The results presented in this paper generalize and improve the corresponding those in the recent literature.

Second-order necessary and sufficient optimality conditions for local solutions and locally unique solutions of generalized quadratic programming problems in Banach spaces are established in this paper. Since the decomposition procedures using orthogonality relations in Euclidean spaces and the compactness of finite-dimensional unit spheres, which worked well for finite-dimensional quadratic programs, cannot be applied to the Banach space setting, a series of new constructions and arguments are proposed. These results give a comprehensive extension of the corresponding theorems on finite-dimensional quadratic programs.

We extend a result on existence of Walrasian equilibria in He and Yannelis (Econ Theory 61:497–513, 2016) by replacing the compactness assumption on consumption sets made there by the standard assumption that these sets are closed and bounded from below. This provides a positive answer to a question explicitly raised in He and Yannelis (Econ Theory 61:497–513, 2016). Our new equilibrium existence theorem generalizes many results in the literature as we do not require any transitivity or completeness or continuity assumption on preferences, initial endowments need not be in the interior of the consumption sets, preferences may be interdependent and price-dependent, and no monotonicity or local non satiation is needed for any of the agents.

We show that the prescribed Gaussian curvature equation in $\mathbb {R}^{2}$ $$ -\varDelta u= (1-|x|^{p}) e^{2u}, $$ has solutions with prescribed total curvature equal to ${{\varLambda }}:={\int \limits }_{\mathbb {R}^{2}}(1-|x|^{p})e^{2u}dx\in \mathbb {R}$ , if and only if $$ p\in(0,2) \qquad \text{and} \qquad (2+p)\pi\le{\varLambda}

In this paper, we provide a constructive proof of $\mathbf {H}^{1}(\mathbb {R}^{n})$ factorization in terms of multilinear Calderon–Zygmund operators in Morrey spaces. As a direct application, we obtain a characterization of functions in $\text {BMO}(\mathbb {R}^{n})$ via commutators of multilinear Calderon–Zygmund operators. Furthermore, we prove a Morrey compactness characterization of [b, T]l, the commutator in the l-th entry.

In this paper we study the principal spectral theory and asynchronous exponential growth for age-structured models with nonlocal diffusion of Neumann type. First, we provide two general sufficient conditions to guarantee existence of the principal eigenvalue of the age-structured operator with nonlocal diffusion. Then we show that such conditions are also enough to ensure that the semigroup generated by solutions of the age-structured model with nonlocal diffusion exhibits asynchronous exponential growth. Compared with previous studies, we prove that the semigroup is essentially compact instead of eventually compact, where the latter is usually obtained by showing the compactness of solution trajectories. Next, following the technique developed in Vo (Principal spectral theory of time-periodic nonlocal dispersal operators of Neumann type. arXiv:1911.06119 , 2019), we overcome the difficulty that the principal eigenvalue of a nonlocal Neumann operator is not monotone with respect to the domain and obtain some limit properties of the principal eigenvalue with respect to the diffusion rate and diffusion range. Finally, we establish the strong maximum principle for the age-structured operator with nonlocal diffusion.

For parameters in a threshold autoregressive process, the paper proposes a sequential modification of the least squares estimates with a specific stopping rule for collecting the data for each parameter. In the case of normal residuals, these estimates are exactly normally distributed in a wide range of unknown parameters. On the base of these estimates, a fixed-size confidence ellipsoid covering true values of parameters with prescribed probability is constructed. In the i.i.d. case with unspecified error distributions, the sequential estimates are asymptotically normally distributed uniformly in parameters belonging to any compact set in the ergodicity parametric region. Small-sample behavior of the estimates is studied via simulation data.

Let $$\mathcal {T}^*$$ and $$\mathcal {I}_\alpha $$ be the semi-group maximal function and fractional integrals associated to the Schrodinger operator $$-\Delta +V(x)$$ , respectively, with V satisfying an appropriate reverse Holder inequality. In this paper, we show that the commutator of $$\mathcal {T}^*$$ is a compact operator on $$L^p(w)$$ for $$1

Let X be a completely regular topological space and f a real-valued bounded from above lower semicontinuous function in it. Let C(X) be the space of all bounded continuous real-valued functions in X endowed with the usual sup-norm. We show that the following two properties are equivalent: X is α-favourable (in the sense of the Banach-Mazur game); The set of functions h in C(X) for which f + h attains its supremum in X contains a dense and Gδ-subset of the space C(X). In particular, property (b) has place if X is a compact space or, more generally, if X is homeomorphic to a dense Gδ subset of a compact space.We show also the equivalence of the following stronger properties: X contains some dense completely metrizable subset; the set of functions h in C(X) for which f + h has strong maximum in X contains a dense and Gδ-subset of the space C(X). If X is a complete metric space and f is bounded, then the set of functions h from C(X) for which f + h has both strong maximum and strong minimum in X contains a dense Gδ-subset of C(X).

In Mergelyan type approximation we uniformly approximate functions on compact sets K by polynomials or rational functions or holomorphic functions on varying open sets containing K. In the present paper we consider analogous approximation, where uniform convergence on K is replaced by uniform approximation on K of all order derivatives.

This paper discusses the solvability and existence of optimal controls of nonautonomous fractional dynamical systems of neutral-type in a general Banach space X along with nonlocal condition. We employ the Banach contraction principle, the classical semigroup theory, and the techniques of fractional calculus to establish the main results. The distinguish features of the presented work are that we establish the main results without imposing the compactness condition on semigroup, the continuity of linear operators $$-A(t)$$ and the strong continuity of operator D. At the end, an example is considered to demonstrate the developed results.

In this paper, a new class of nonexpansive mappings is defined and some fixed point theorems for such newly type are proved in the setting of bounded metric spaces without using neither the compactness nor the so-called uniform convexity. Our theorems generalize and improve many known results in the fixed point theory. Furthermore, we apply the main results to show the existence and uniqueness of a solution for a differential equation.

We establish the existence and multiplicity of solutions for a Kirchhoff-type problem in $$\mathbb R^4$$ involving a critical and concave–convex nonlinearity. Since in dimension four, the Sobolev critical exponent is $$2^*=4$$ , there is a tie between the growth of the nonlocal term and the critical nonlinearity. This turns out to be a challenge to study our problem from the variational point of view. Some of the main tools used in this paper are the mountain-pass and Ekeland’s theorems, Lions’ Concentration Compactness Principle and an extension to $$\mathbb R^N$$ of the Struwe’s global compactness theorem.

We investigate the relations between Pesin-Pitskel topological pressure on an arbitrary subset and measure-theoretic pressure of Borel probability measures for finitely generated semigroup actions. Let (X, $${\cal G}$$ ) be a system, where X is a compact metric space and $${\cal G}$$ is a finite family of continuous maps on X. Given a continuous function f on X, we define Pesin-Pitskel topological pressure $${P_{\cal G}}(Z,f)$$ for any subset Z ⊂ X and measure-theoretical pressure $${P_{\mu ,{\cal G}}}(X,f)$$ for any $$\mu \in {\cal M}(X)$$ , where $${\cal M}(X)$$ denotes the set of all Borel probability measures on X. For any non-empty compact subset Z of X, we show that $${P_{\cal G}}(Z,f) = \sup \{ {P_{\mu ,{\cal G}}}(X,f):\mu \in {\cal M}(X),\mu (Z) = 1\} .$$

For two analytic self-maps φ and ψ defined on the unit disk ⅅ, we characterize completely the boundedness and compactness of the difference Cφ − Cψ of the composition operators Cφ and Cψ from Bloch space B into Besov space $$B_\nu ^\infty $$ . Moreover, we also give a complete characterization of the compactness of the difference Cφ − Cψ on BMOA space.

By employing the notion of exceptional family of elements, we establish existence results for the mixed tensor variational inequalities. We show that the nonexistence of an exceptional family of elements is a sufficient condition for the solvability of mixed tensor variational inequality. For positive semidefinite mixed tensor variational inequalities, the nonexistence of an exceptional family of elements is proved to be an equivalent characterization of the nonemptiness of the solution sets. We derive several sufficient conditions of the nonemptiness and compactness of the solution sets for the mixed tensor variational inequalities with some special structured tensors. Finally, we show that the mixed tensor variational inequalities can be defined as a class of convex optimization problems.

In present work, we have studied a new stellar distribution model with spherically symmetric matter and an uncharged isotropic distribution in general relativity. In this model, we have considered a particular metric potential. The model is capable to represents some known compact stars like Her X-1,4U 1538-52 and SAX J1808.4-3658.The model satisfy the energy condition and hydrostatic equilibrium equation, i.e., the modified Tolman-Oppenheimer-Volkoff (TOV) equation for uncharged matter. In addition to this, we also present the velocity of sound, surface redshift, and pressure density ratio. The physical quantities such as pressure, density, redshift, etc., are compared with graphical representations that are important from theoretical and astrophysical scale.

We study Toeplitz operators from Hardy spaces to weighted Bergman spaces in the unit ball of Cn. Toeplitz operators are closely related to many classical mappings, such as composition operators, the Volterra type integration operators and Carleson embeddings. We characterize the boundedness and compactness of Toeplitz operators from Hardy spaces Hp to weighted Bergman spaces A for the different values of p and q in the unit ball.

In this paper, we present various sufficient conditions for a family of meromorphic mappings on a domain $$D\subset {\mathbb {C}}^m$$ into $${\mathbb {P}}^n$$ to be meromorphically normal. Meromorphic normality is a notion of sequential compactness in the meromorphic category introduced by Fujimoto. We give a general condition for meromorphic normality that is influenced by Fujimoto’s work. The approach to proving this result allows us to establish meromorphic analogues of several recent results on normal families of $${\mathbb {P}}^n$$ -valued holomorphic mappings.

The Whitham equation is a nonlocal shallow water-wave model which combines the quadratic nonlinearity of the KdV equation with the linear dispersion of the full water wave problem. Whitham conjectured the existence of a highest, cusped, traveling-wave solution, and his conjecture was recently verified in the periodic case by Ehrnstr\"om and Wahl\'en. In the present paper we prove it for solitary waves. Like in the periodic case, the proof is based on global bifurcation theory but with several new challenges. In particular, the small-amplitude limit is singular and cannot be handled using regular bifurcation theory. Instead we use an approach based on a nonlocal version of the center manifold theorem. In the large-amplitude theory a new challenge is a possible loss of compactness, which we rule out using qualitative properties of the equation. The highest wave is found as a limit point of the global bifurcation curve., Comment: Journal version. Mathematische Annalen, Online First. 45 pages, 3 figures

This paper introduces a ‘new’ class of operators. An operator T from a Banach lattice E into a Banach space Y will be called an LW-compact if T carries L-weakly compact sets of E into relatively compact sets of Y. Various characterizations of LW-compact operators are given. Also, we study the domination problem in this class of operators.

We give a new criterion for compactness of the multiplication operator $$M_u$$ acting on $$L_p(\mu )$$ spaces in terms of the atomic and non-atomic decomposition. As an important consequence, we also obtain an estimation of the essential norm of $$M_u$$ acting on $$L_p(\mu )$$ spaces in the more general setting.

In this paper, we consider robust output tracking for an Euler–Bernoulli beam equation under the guidance of the internal model principle, where the disturbances in all possible channels are considered. Three typical cases are investigated in terms of different regulated outputs. The first case is based on boundary displacement output, for which only asymptotic convergence can be achieved due to the compactness of the observation operator. The second case considers two outputs of both boundary displacement and velocity. Since the control is one-dimensional, we can only arbitrarily regulate the boundary displacement and at the same time, the velocity is regulated to track the derivative of the reference. This is not the standard form investigated in the literature for robust error feedback control of abstract infinite-dimensional systems. The last case represents an extreme case that the system is non-well posed. In all the above cases, this paper demonstrates the same technique of an observer-based approach to robust control design. In the latter two cases, we can achieve exponential convergence and the closed loop is also shown to be robust to system uncertainties. Numerical simulations are carried out in all cases to illustrate the effectiveness of the proposed controls.

In this paper, we consider a class of fourth degree polynomial problems, which are NP-hard. First, we are concerned with the bi-quadratic optimization problem (Bi-QOP) over compact sets, which is proven to be equivalent to a multi-linear optimization problem (MOP) when the objective function of Bi-QOP is concave. Then, we introduce an augmented Bi-QOP (which can also be regarded as a regularized Bi-QOP) for the purpose to guarantee the concavity of the underlying objective function. Theoretically, both the augmented Bi-QOP and the original problem share the same optimal solutions when the compact sets are specified as unit spheres. By exploiting the multi-block structure of the resulting MOP, we accordingly propose a proximal alternating minimization algorithm to get an approximate optimal value of the problem under consideration. Convergence of the proposed algorithm is established under mild conditions. Finally, some preliminary computational results on synthetic datasets are reported to show the efficiency of the proposed algorithm.

We study the compactness of composition operators on the Bergman spaces of certain bounded convex domains in $\mathbb{C}^n$ with non-trivial analytic discs contained in the boundary. As a consequence we characterize that compactness of the composition operator with a continuous symbol (up to the closure) on the Bergman space of the polydisc., To appear in Analysis Mathematica, 9 pages. arXiv admin note: substantial text overlap with arXiv:2006.06498

In this paper we generalise the results proved in [N. Katzourakis, An $L^\infty$ regularisation strategy to the inverse source identification problem for elliptic equations, SIAM J. Math. Anal. 51:2, 1349-1370 (2019)] by studying the ill-posed problem of identifying the source of a fully nonlinear elliptic equation. We assume Dirichlet data and some partial noisy information for the solution on a compact set through a fully nonlinear observation operator. We deal with the highly nonlinear nonconvex nature of the problem and the lack of weak continuity by introducing a two-parameter Tykhonov regularisation with a higher order $L^2$ "viscosity term" for the $L^\infty$ minimisation problem which allows to approximate by weakly lower semicontinuous cost functionals., 14 pages. arXiv admin note: text overlap with arXiv:1811.02845

We provide a constructive proof of H1 (ℝd) (the classical Hardy space) factorization in terms of fractional commutators in Lorentz spaces. As a direct application, we obtain a characterization of functions in BMO space. Furthermore, we also obtain a Lorentz compactness characterization of fractional commutators.

We give two characterizations, one for the class of generalized Young measures generated by $\mathcal A$-free measures, and one for the class generated by $\mathcal B$-gradient measures $\mathcal Bu$. Here, $\mathcal A$ and $\mathcal B$ are linear homogeneous operators of arbitrary order, which we assume satisfy the constant rank property. The characterization places the class of generalized $\mathcal A$-free Young measures in duality with the class of $\mathcal A$-quasiconvex integrands by means of a well-known Hahn--Banach separation property. A similar statement holds for generalized $\mathcal B$-gradient Young measures. Concerning applications, we discuss several examples that showcase the rigidity or the failure of $\mathrm{L}^1$-compensated compactness when concentration of mass is allowed. These include the failure of $\mathrm{L}^1$-estimates for elliptic systems and the failure of $\mathrm{L}^1$-rigidity for the two-state problem. As a byproduct of our techniques we also show that, for any bounded open set $\Omega$, the inclusions \[ \mathrm{L}^1(\Omega) \cap \ker \mathcal A \hookrightarrow \mathcal M(\Omega) \cap \ker \mathcal A, \] \[ \{\mathcal B u\in \mathrm{C}^\infty(\Omega)\} \hookrightarrow \{\mathcal B u\in \mathcal M(\Omega)\}, \] are dense with respect to area-functional convergence of measures, Comment: 73 pages, 3 figures. Version 4 (accepted for publication in Arch. Ration. Mech. Anal.) incorporates the characterization of $\mathcal B$-gradient measures, several new examples and several new applications that discuss the failure of $L^1$ compensated compactness for elliptic systems and the $n$-state problem

The notion of the (dual) hull-kernel topology on a collection of prime filters in a residuated lattice is introduced and investigated. It is observed that any collection of prime filters is a $$T_0$$ topological space under the (dual) hull-kernel topology. It is proved that any collection of prime filters is a $$T_1$$ space if and only if it is an antichain, and it is a Hausdorff space if and only if it satisfies some certain conditions. Some characterizations in which maximal filters form a Hausdorff space are given. In the end, we focus on the space of minimal prim filters, and verify that this space is totally disconnected Hausdorff. This paper is closed by description of the compactness of the space of the minimal prime filters using the space of prime $$\alpha $$ -filters.

Pollution management models of Shallow Lake type identify a well-known class of infinite horizon optimal control problems, characterized by the absence of concavity in the state equation and of compactness in the control space. The seminal paper by Maler et al. (2003) generated a consistent stream of literature over the years, even though existence of solutions to the optimization problem had for long remained an open question. In a recent paper by the author, an optimal policy has been proven to exist, for models describing an endogenous pollution dynamics that decreases globally with the total amount of pollution. The present paper is concerned with the complementary situation, the one in which, mathematically speaking, the velocity field changes its monotonicity with respect to the space variable. For Shallow Lake models, such property corresponds to an hysteresis phenomenon. We prove the existence of an optimum in the same class as in the last-mentioned article, under the assumption that the discount exponent in the objective functional is sufficiently big compared to the spatial derivative of the velocity field. From the methodological viewpoint, this goal requires a significant improvement of the technique introduced to solve the monotonic problem.

We introduce a connection between Newhouse thickness and patterns through a variant of Schmidt’s game introduced by Broderick, Fishman and Simmons. This yields an explicit, robust and checkable condition that ensures that a Cantor set in the real line contains long arithmetic progressions and, more generally, homothetic copies of large finite sets.

This paper is devoted to investigating the Freidlin–Wentzell’s large deviation principle for a class of McKean–Vlasov quasilinear SPDEs perturbed by small multiplicative noise. We adopt the variational framework and the modified weak convergence criteria to prove the Laplace principle for McKean–Vlasov type SPDEs, which is equivalent to the large deviation principle. Moreover, we do not assume any compactness condition of embedding in the Gelfand triple to handle both the cases of bounded and unbounded domains in applications. The main results can be applied to various McKean–Vlasov type SPDEs such as distribution dependent stochastic porous media type equations and stochastic p-Laplace type equations.

In this paper, we characterize that the boundedness, compactness and spectral structure of dual Toeplitz operators acting on the orthogonal complement of the harmonic Bergman space. This generalizes the corresponding results for dual Toeplitz operators on the orthogonal complement of the Bergman space due to Stroethoff and Zheng’s paper [Trans. Amer. Math. Soc., 354, 2495–2520 (2002)].

This paper investigates the nonemptiness and compactness of the mild solution set for a class of Riemann-Liouville fractional delay differential variational inequalities, which are formulated by a Riemann-Liouville fractional delay evolution equation and a variational inequality. Our approach is based on the resolvent technique and a generalization of strongly continuous semigroups combined with Schauder’s fixed point theorem.

In this paper, we present a new exact anisotropic charged version of Pant’s interior solution [Astrophys. Space Sci. 331, 633 (2011)] by employing the embedding class I condition and the gravitational decoupling via minimal geometric decoupling (MGD) approach. We successfully integrate the coupled Einstein–Maxwell field equations within the MGD and generate a family of new charged anisotropic solutions describing compact stellar objects. The matching of the interior to the exterior Reissner–Nordstrom solution help fix the arbitrary constants. For a chosen parameter space we subject our model to rigorous tests in order to validate its physical viability as a realistic self-gravitating, compact object. We show that the deformation parameter $$\sigma$$ is intrinsically connected to key features such as compactness, mass-to-radius ratio and surface redshift for compact star candidates such as Her X-1, EXO 1785-248, PSR J1614-2230 and SAX J1808.4-3658.

In this paper, we study the boundedness and compactness of the Volterra integral operators from Dirichlet type spaces $${\mathcal {D}}^p_{p-2+s}$$ into Hardy spaces $${\mathcal {H}}^q$$ when $$1

The boundedness and compactness of embedding from Dirichlet-Morrey spaces $${\mathcal {D}}_{p}^{\lambda }$$ into tent spaces $${\mathcal {T}}^q_{ s}(\mu )$$ are studied in this paper. For an application, a characterization for the boundedness of the Volterra operator $$T_g$$ from $${\mathcal {D}}_{p}^{\lambda }$$ to $$F\left( q,\frac{qp(1-\lambda )}{2},s\right) $$ is given. The essential norm and compactness of $$T_g$$ from $${\mathcal {D}}_{p}^{\lambda }$$ to $$F\left( q,\frac{qp(1-\lambda )}{2},s\right) $$ are also investigated.

In this paper a model of cell proliferation with unequal division is analyzed. The model uses an integral equation to describe the dynamics of the cells density. The analysis of the model is done using tools of the theory developed for the class of perturbed dual semigroups that are solutions of an abstract integral equation. Existence, uniqueness, positivity, compactness and spectral properties of the solution semigroup are derived in order to conclude the asynchronous exponential growth property for the model.

Let $u \in C^{1}(\mathbb {R}^{N})$ be a weak solution to the equation $$ -{{\varDelta}}_{p} u = \left(\frac{1}{|x|^{N-\alpha}} \ast |u|^{q}\right) |u|^{q-2}u \quad\text{ in } \mathbb{R}^{N}, $$ where 2 ≤ p < N and $\max \limits \{0,N-2p\}

Let $${{\mathcal {A}}}$$ be a Banach algebra and let $$\varphi $$ be a non-zero character on $${{\mathcal {A}}}$$ . Suppose that $${{\mathcal {A}}}_M$$ is the closure of the faithful Banach algebra $${{\mathcal {A}}}$$ in the multiplier norm. In this paper, topologically left invariant $$\varphi $$ -means on $${{\mathcal {A}}}_M^*$$ are defined and studied. Under some conditions on $${{\mathcal {A}}}$$ , we will show that the set of topologically left invariant $$\varphi $$ -means on $${{\mathcal {A}}}^*$$ and on $${{\mathcal {A}}}_M^*$$ have the same cardinality. The main applications are concerned with the quantum group algebra $$L^1({\mathbb {G}})$$ of a locally compact quantum group $${\mathbb {G}}$$ . In particular, we obtain some characterizations of compactness of $${\mathbb {G}}$$ in terms of the existence of a non-zero (weakly) compact left or right multiplier on $$L^1_M({\mathbb {G}})$$ or on its bidual in some senses.

This paper develops a unified method to derive compactness properties and convergence to a steady state as well as decay estimates of global bounded solutions of the abstract semilinear second order integro-differential equation $$\begin{aligned} u_{tt}(t)+ Au(t) -\int ^\tau _0 k(s) Au(t-s)ds+ f(u)=g(t), \ \tau \in \{t, \infty \}, \end{aligned}$$ with finite ( $$\tau =t$$ ) or infinite ( $$\tau =\infty $$ ) memory. Here, A is a continuous linear operator, k is a positive nonincreasing function, f is a nonlinear function such that $$A+f$$ is the gradient of a potential satisfying the Łojasiewicz–Simon inequality, and the time-dependent right-hand side g decays to 0 at infinity. The general results are applied to particular semilinear evolution viscoelastic problems at the end of the paper.

In this paper, we extend compactness theorems of Cheeger, Gromov, Taylor, and Sprouse to the Bakry–Emery Ricci tensor and generalized quasi-Einstein tensors. Our results generalize previous results obtained by Yun and Wan.

In this paper, we introduce the concept of measure theoretic r-entropy on a compact metric space in the context of amenable group actions. To be precise, we generalize Katok’s r-entropy formula and Brin-Katok r-entropy formula to infinite countably amenable group actions.

It is well-known that the Kadec-Klee property is an important property in the geometry of Banach spaces. It is closely connected with the approximation compactness and fixed point property of non-expansive mappings. In this paper, a criterion for Musielak-Orlicz function spaces equipped with the Orlicz norm to have the Kadec-Klee property are given. As a corollary, we obtain that a class of non-reflexive Musielak-Orlicz function spaces have the Fixed Point property.

In this paper, we consider the following new Kirchhoff-type equations involving the fractional p-Laplacian and Hardy-Littlewood-Sobolev critical nonlinearity: $$\begin{array}{*{20}{c}} {{{\left( {a + b\iint_{{\mathbb{R}^N}} {\frac{{{{\left| {u(x) - u(y)} \right|}^p}}}{{{{\left| {x - y} \right|}^{N + ps}}}}\text{d}x\text{d}y}} \right)}^{p - 1}}}&{( - \Delta )_p^su + \lambda V(x){{\left| u \right|}^{p - 2}}u} \\ { = \left( {\int\limits_{{\mathbb{R}^N}} {\frac{{{{\left| u \right|}^{p_{\mu ,s}^*}}}}{{{{\left| {x - y} \right|}^\mu }}}\text{d}y} } \right){{\left| u \right|}^{p_{\mu ,{s^{ - 2}}}^*}}u,\;x \in {\mathbb{R}^N}}&\; \end{array}\;$$ where (−Δ) is the fractional p-Laplacian with 0 ps, a, b > 0, λ > 0 is a parameter, V: ℝN → ℝ+ is a potential function, θ ∈ [1, 2 μ,s * ) and $$^{p_{\mu,s}^ * = {{pN - p{\mu \over 2}} \over {N - ps}}}$$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. We get the existence of infinitely many solutions for the above problem by using the concentration compactness principle and Krasnoselskii’s genus theory. To the best of our knowledge, our result is new even in Choquard-Kirchhoff-type equations involving the p-Laplacian case.

We introduce the notion of $$q^\prime $$ -compactness for Sheffer stroke basic algebras and Visser algebras. Our goal is to determine when induced lattice of a Sheffer stroke basic algebra and a Visser algebra is a strongly algebraically closed algebra, and we find the condition that the lattices of complete congruences relations on a Sheffer stroke basic algebra are weakly relatively pseudocomplemented. In particular, an open question proposed by A. Di-Nola, G. Georgescu and A. Iorgulescu about the connections of dually Brouwerian pseudo-BL-algebras with other algebraic structures in Di Nola et al. (Mult Val Logic 8:717–750, 2002) is answered.

An interesting observation is that most pairs of weakly homogeneous mappings do not possess strongly monotonic property, which is one of the key conditions to ensure the unique solvability of the generalized variational inequality. This paper focuses on studying the uniqueness and solvability of the generalized variational inequality with a pair of weakly homogeneous mappings. By using a weaker condition than the strong monotonicity and some additional conditions, we achieve several results on the unique solvability to the underlying problem, which are exported by making use of the exceptional family of elements. As an adjunct, we also obtain the nonemptiness and compactness of the solution sets to the weakly homogeneous generalized variational inequality under some appropriate conditions. The conclusions presented in this paper are new or supplements to the existing ones even when the problem comes down to its important subclasses studied in recent years.

We show that the tree property, stationary reflection and the failure of approachability at $$\kappa ^{++}$$ are consistent with $$\mathfrak {u}(\kappa )= \kappa ^+ < 2^\kappa $$ , where $$\kappa $$ is a singular strong limit cardinal with the countable or uncountable cofinality. As a by-product, we show that if $$\lambda $$ is a regular cardinal, then stationary reflection at $$\lambda ^+$$ is indestructible under all $$\lambda $$ -cc forcings (out of general interest, we also state a related result for the preservation of club stationary reflection).

We give a perhaps less known version of Hartogs’ extension for connected, non-compact complex manifolds M such that $$H^1_c(M,{\mathcal{{O}}})=0$$ . Then we prove extensions/non-extensions for similar questions related to holomorphic vector/line bundles over complements of Stein compact sets in Stein manifolds.

In this paper we propose a definition of a distance between flows of possibly different metric spaces. This notion is then combined with the concept of topologically stable flow to obtain the notion of $$\sigma $$ -topologically GH-stable flow. Afterwards, we prove that an expansive flow of a compact metric space, with the pseudo-orbit tracing property, and without singularities is $$\sigma $$ -topologically GH-stable.